In section14I reviewed the basics for calculations in a1 + 1dimensional quantum field theory, which is invariant under Galilean transformations in addition to having scale invariance. I will now elaborate on how the results obtained previously are related to flat space holography in2 + 1dimensions.
Taking this BMS/GCA correspondence [65] into account all the field theory results presented earlier in this chapter in the context of GCFT can be used for calculations for ultrarelativistic field theories invariant underbms3 symmetries at null infinity of asymptotically flat spacetimes by only exchanging the role of time and space.
Using this argument one can immediately determine entanglement entropy in a planar field theory having ultrarelativistic conformal/bms3 symmetry. Once again assuming a rectilinear segmentAwith end points(t1, x1)and(t2, x2)as the entan-gling region, one can readily determine the entanglement entropy of that region by using (14.26) and exchanging time and space
SE = cL
This interval can again be interpreted as a boosted version of a purely spatial (or equal time) interval. Fort12= 0(14.27) reduces to
SE = cL
The entanglement entropy calculated above corresponds to a 1 + 1 dimensional system of infinite spatial extent at zero temperature. It is also of interest to see what happens with the entanglement entropy when dealing with a system at finite temperatureT =β−1 and/or finite spatial extent.
This generalization can be achieved by using geometric properties of the 1 + 1 dimensional field theory in question very much alike to the CFT case, where one can use conformal maps to map between the entanglement entropy of different systems.
To elaborate on this further, note that one can map the1 + 1dimensional GCFT on the plane to a cylinder by
x=e2πξ/β, t= 2πτ
β e2πξ/β, (14.29)
whereξ andτ denote the coordinates on the cylinder. This means effectively that one dimension gets compactified in the construction ofΣn. As shown in [71] this induces a transformation of the GCFT primaries as
Φ(ξ, τ˜ ) =e2πβ (ξhL+τ hM)Φ(x(ξ, τ), t(ξ, τ)). (14.30) The two point function evaluated in this geometry is then given by
hΦ(ξ˜ 1, τ1) ˜Φ(ξ2, τ2)i=
The following steps which are necessary for calculating entanglement entropy in a thermal state for a subsystem with endpointsξ1, τ1andξ2, τ2 are the same as in the zero-temperature case. Thus one obtains for the entanglement entropy for a system at finite temperature the following expression
SE = cL
At leading order the expansion of the right hand side of (14.32) inβ−1yields again the zero temperature answer (14.27) with the identification of τ12 ∼ t12/a and ξ12∼x12/a. In the high temperature limit on the other hand, i.e. forξ12β, one obtains
SE = π
6β(cLξ12+cMτ12) +cL
6 lnβ+O(β). (14.33) A very similar analysis works when considering the spatial extent of the system to be of finite lengthLin the ground state. The only difference in comparison to the analysis before lies in the direction of the compactification to the cylinder along the spatial cycle of lengthL∼β which is perpendicular to the previous case. The entanglement entropy for that system then turns out to be
SE = cL
14.5 Thermal Entropy in GCFTs
In this section I will briefly review how to derive the high-temperature density of states and the corresponding entropy for ordinary 1 + 1dimensional GCFTs (for
more details see e.g. [61,62,71]) in order to make contact with the holographic results for the thermal entropy of FSCs found in section15.4.
The partition function for a1 + 1dimensional GCFT on a torus is given by ZGCF T0 (η, ρ) =Tre2πiη(L0−cL
24)e2πiρ(M0−cM
24)
=e12πi(ηcL+ρcM)ZGCF T(η, ρ), (14.35) whereηandρare the Galilean conformal equivalents of the modular parameters of a CFT. In the same spirit as in a relativistic CFT one demands that (14.35) is invariant under the Galilean conformal equivalent ofSmodular transformations given by
(η, ρ)→
This is tantamount to requiring
ZGCF T(η, ρ) =e2πi(f(η,ρ)+h˜ Lη+hMρ)ZGCF T(−1 In order proceed, one has first to rewrite the density of statesd(hL, hM)in terms of the GCA partition function. This can be done by using
ZGCF T(η, ρ) =Tre2πiηL0e2πiρM0=Xd(hL, hM)e2πiηhLe2πiρhM, (14.40) and performing an inverse Laplace transformation
d(hL, hM) = Z
dηdρe2πif(η,ρ)˜ ZGCF T(−1 η, ρ
η2). (14.41) In the limit of large central charges the density of states (14.41) can by approximated by the value of the integrand, when the exponential factor is extremal. Using this approximation the density of states is given by
d(hL, hM)∼eπ
The corresponding entropy is then given by the logarithm of the density of states S= ln (d(hL, hM)) =π
For Einstein gravity, where cL = 0 the entropy (14.43) agrees precisely with the thermal entropy of a flat space cosmology [61,62]. Thus, the 2d GCFT state counting reproduces exactly the entropy of the cosmological horizon of a FSC. This is one particular example of a specific check that one can indeed establish a holographic principle for asymptotically flat spacetimes.
14.6 Early Checks of Flat Space Holography
Further selected checks, some of which I also mentioned briefly in the introduction include:
Barnich and Compère proposed in 2006 [51] asymptotic boundary conditions for flat spacetimes in three dimensional Einstein gravity and showed that the asymptotic symmetries corresponding to these boundary conditions are given by thebms3 algebra (10.3) with central chargescL= 0andcM = G3
N.
Bagchi then showed in 2010 [56] (and also together with Fareghbal in [65]) that thebms3 algebra is isomorphic to the Galilean conformal algebragca2, which he coined the BMS/GCA correspondence, and thus was able to propose a framework for the dual field theory of asymptotically flat spacetimes.
Even though the framework for a dual theory was found in [56,65], a concrete proposal for a specific theory withcL= 0andcM 6= 0was missing. ForcL6= 0 and cM = 0, however, the situation was the complete opposite. The field theory dual was clear, as it would be just a chiral half of a CFT, whereas the corresponding gravity dual was unknown. In [57] Bagchi, Detournay and Grumiller found the gravity dual whose asymptotic symmetry algebra isbms3 withcL6= 0andcM = 0. This theory is known as flat space chiral gravity and can be obtained as a flat space limit from Topologically Massive Gravity [152]
whose action is given by (10.49).
In 2013 Bagchi et al. showed that there exist Hawking-Page like phase transi-tions between flat space cosmologies and hot flat space3at a critical tempera-tureTc= 2πr1
0 , wherer0is the radius of the cosmological horizon.
All these explicit checks, and many more that I did not explicitly mention, strongly suggest that doing flat space holography is indeed possible.
In the next chapter I will provide an additional explicit check of flat space holography by computing the entanglement entropy of abms invariant quantum field theory holographically.
3This is basically the Euclidean version of the null orbifold.